Author Archives: Jason Polak

Let's suppose $M$ is a nonzero projective $\Z$-module. Can it be finite? Nope. I'm sure there are plenty ways to prove it, but one way is to observe that a projective $\Z$-module is free, and hence if $M$ is nonzero it must have at least one copy of $\Z$. So, $M$ is infinite. What's the […]

This is mostly a continuation on the group I gave in the last post, which is given by the presentation $$G = \langle a,t ~|~ t^{-1}a^2t = a^3\rangle.$$ At the risk of beating a dead horse, I proved that the homomorphism $f:G\to G$ given on generators by $f(t) = t$ and $f(a) = a^2$ is […]

A few weeks ago I gave an example of a non-Hopfian finitely-presented group. Recall that a group $G$ is said to be Hopfian if every surjective group homomorphism $G\to G$ is actually an isomorphism. All finitely-generated, residually finite groups are Hopfian. So for example, the group of the integers $\Z$ is Hopfian. Another example of […]

An associative ring $R$ is called von Neumann regular if for each $x\in R$ there exists a $y\in R$ such that $x = xyx$. Now let $R$ be a commutative ring. Its dimension is the supremum over lengths of chains of prime ideals in $R$. So for example, fields are zero dimensional because the only […]

Let's see an example of a finitely-generated flat module that is not projective! What does this provide a counterexample to? If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module […]

Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved […]

In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free […]

This is a meta post on blogging, not mathematics. Recently, I got it into my head that I should design my own WordPress theme from scratch. As a consequence, you may have noticed that the theme of this blog has changed a little. I don't know if many other math bloggers will want to try […]

When it comes to math humour, there's not much out there. There is a good list of jokes on MathOverflow. There's also Mathematical Apocrypha by Krantz, many of whose folklore stories are also amusing. The other day at the library I found another one: Riot at the Calc Exam by Colin Adams. Adams' book is […]